116
Analytical Heat Transfer
p step
1
2
h, T ∞
x
y
p + 1 step
Δx
p+1
T 1
p+1
T 0
p
T 1
p
T 0
p
T 2
p
T 3
T i
p+1
T 2
p+1
T 3
FIGURE 5.8
Finite difference energy balance method for one-dimensional transient heat conduction with
convection boundary conditions.
Example 5.5
For the case of surface heat flux BC as sketched in Figure 5.9:
T
p+1
�
�
T
p
− T
p
1 − T
p
Δx
""
0
0
0
q s + k
=
ρC p
(5.18)
Δx
Δt
2
T
p+1
Δt
""
k Δt
T
p
=
q +
1 + (1 − 2Fo)T
p
(5.19)
0
s
0
(Δx/2)ρC p
(Δx/2)ρC p
For stability, Fo ≤ 1/2
1
2
Δx
p+1
T 3
p
T 3
T i
p
T 2
p
T 1
p
T 0
p+1
T 2
p+1
T 1
p+1
T 0
p + 1 step
p step
q" s
x
y
FIGURE 5.9
Finite difference energy balance method for 1-D transient heat conduction with surface heat flux
boundary condition.
Analytical Heat Transfer
p step
1
2
h, T ∞
x
y
p + 1 step
Δx
p+1
T 1
p+1
T 0
p
T 1
p
T 0
p
T 2
p
T 3
T i
p+1
T 2
p+1
T 3
FIGURE 5.8
Finite difference energy balance method for one-dimensional transient heat conduction with
convection boundary conditions.
Example 5.5
For the case of surface heat flux BC as sketched in Figure 5.9:
T
p+1
�
�
T
p
− T
p
1 − T
p
Δx
""
0
0
0
q s + k
=
ρC p
(5.18)
Δx
Δt
2
T
p+1
Δt
""
k Δt
T
p
=
q +
1 + (1 − 2Fo)T
p
(5.19)
0
s
0
(Δx/2)ρC p
(Δx/2)ρC p
For stability, Fo ≤ 1/2
1
2
Δx
p+1
T 3
p
T 3
T i
p
T 2
p
T 1
p
T 0
p+1
T 2
p+1
T 1
p+1
T 0
p + 1 step
p step
q" s
x
y
FIGURE 5.9
Finite difference energy balance method for 1-D transient heat conduction with surface heat flux
boundary condition.